April  2000, 6(2): 393-418. doi: 10.3934/dcds.2000.6.393

Uniform inertial sets for damped wave equations

1. 

Université de Bordeaux I, Laboratoire de Mathématiques Appliquées de Bordeaux, 351 cours de la libération, 33400 Talence, France

2. 

Université de Bordeaux I, Laboratoire de Mathématiques Appliquées de Bordeaux, 351 Cours de la Libération, 33405 Talence Cedex, France

3. 

Université de Poitiers, Département de Mathématiques, 40 Avenue du Recteur Pineau, 86022 Poitiers Cedex, France

Received  March 1999 Revised  June 1999 Published  January 2000

In this paper, we establish the existence of inertial sets for a class of wave equations in which the coefficient of the second order time derivative is $\varepsilon$. We show that the fractal dimension of these inertial sets does not depend on $\varepsilon$ for $\varepsilon$ small enough. We then compare the asymptotic behavior of the problem (as $\varepsilon\to 0$) through a continuity like property of the inertial sets. The autonomous case and nonautonomous case are studied.
Citation: P. Fabrie, C. Galusinski, A. Miranville. Uniform inertial sets for damped wave equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 393-418. doi: 10.3934/dcds.2000.6.393
[1]

Pierre Fabrie, Alain Miranville. Exponential attractors for nonautonomous first-order evolution equations. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 225-240. doi: 10.3934/dcds.1998.4.225

[2]

Gisèle Ruiz Goldstein, Jerome A. Goldstein, Fabiana Travessini De Cezaro. Equipartition of energy for nonautonomous wave equations. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 75-85. doi: 10.3934/dcdss.2017004

[3]

Ioana Moise, Ricardo Rosa, Xiaoming Wang. Attractors for noncompact nonautonomous systems via energy equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 473-496. doi: 10.3934/dcds.2004.10.473

[4]

Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 579-596. doi: 10.3934/dcds.2006.15.579

[5]

Yonghai Wang, Chengkui Zhong. Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3189-3209. doi: 10.3934/dcds.2013.33.3189

[6]

Marcello D'Abbicco, Giovanni Girardi, Giséle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Equipartition of energy for nonautonomous damped wave equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020364

[7]

A. Rodríguez-Bernal. Perturbation of the exponential type of linear nonautonomous parabolic equations and applications to nonlinear equations. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1003-1032. doi: 10.3934/dcds.2009.25.1003

[8]

Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023

[9]

Sana Netchaoui, Mohamed Ali Hammami, Tomás Caraballo. Pullback exponential attractors for differential equations with delay. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020367

[10]

Mohamed Ali Hammami, Lassaad Mchiri, Sana Netchaoui, Stefanie Sonner. Pullback exponential attractors for differential equations with variable delays. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 301-319. doi: 10.3934/dcdsb.2019183

[11]

Valeria Danese, Pelin G. Geredeli, Vittorino Pata. Exponential attractors for abstract equations with memory and applications to viscoelasticity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2881-2904. doi: 10.3934/dcds.2015.35.2881

[12]

Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169

[13]

Pierre Fabrie, Cedric Galusinski, A. Miranville, Sergey Zelik. Uniform exponential attractors for a singularly perturbed damped wave equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 211-238. doi: 10.3934/dcds.2004.10.211

[14]

Jianhua Huang, Wenxian Shen. Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 855-882. doi: 10.3934/dcds.2009.24.855

[15]

Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020206

[16]

Dung Le. Exponential attractors for a chemotaxis growth system on domains of arbitrary dimension. Conference Publications, 2003, 2003 (Special) : 536-543. doi: 10.3934/proc.2003.2003.536

[17]

John M. Ball. Global attractors for damped semilinear wave equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 31-52. doi: 10.3934/dcds.2004.10.31

[18]

Bixiang Wang, Xiaoling Gao. Random attractors for wave equations on unbounded domains. Conference Publications, 2009, 2009 (Special) : 800-809. doi: 10.3934/proc.2009.2009.800

[19]

Radosław Kurek, Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1955-1981. doi: 10.3934/dcds.2018079

[20]

Peter E. Kloeden, José Real, Chunyou Sun. Robust exponential attractors for non-autonomous equations with memory. Communications on Pure & Applied Analysis, 2011, 10 (3) : 885-915. doi: 10.3934/cpaa.2011.10.885

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (36)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]